Proving that a vector field is conservative

[JD_PM]

Homework Statement

Homework Equations

$$F = \nabla \phi$$

The Attempt at a Solution

Let's focus on determining why this vector field is conservative. The answer is the following:

[/B]
I get everything till it starts playing with the constant of integration once the straightforward differential equations have been solved.

May you explain how does it conclude that $\phi (x, y, z)$ is a potential for $F$?

Thanks.

 

[DrClaude]

You want to find $\phi$ by integrating $F$. You start by integrating with respect to $x$, which tells you about the function form of $\phi$ with respect to $x$, up to a constant $C_1$, which is a constant wrt $x$ but can be a function of $y$ and $z$. You then find how $C_1$ changes with $y$ by integrating over $y$, and so on.

 

[JD_PM]

You want to find $\phi$ by integrating $F$. You start by integrating with respect to $x$, which tells you about the function form of $\phi$ with respect to $x$, up to a constant $C_1$, which is a constant wrt $x$ but can be a function of $y$ and $z$. You then find how $C_1$ changes with $y$ by integrating over $y$, and so on.

But I do not understand why $\frac{\partial \phi}{\partial y} = \frac{\partial C_1}{\partial y}$

 

[vela]

At that point in the solution, what is $\phi$ equal to? What do you get when you differentiate it with respect to $y$?